## Tuesday, April 22, 2014

### Lattice Rationals, 7 of 10

Lattice rational arithmetic; definitions and laws

Define addition and subtraction by the compensator rule:
a/b + c/d     =   ( a*(d;b) + c*(b;d)  )  /  lcm(b,d)
a/b - c/d     =    ( a*(d;b) – c*(b;d)  )   /  lcm(b,d)

Define multiplication, reciprocal and division the usual ways:
(a/b)*(c/d)    =   (a*c) / (b*d)
1/(c/d)    =    (d/c)
(a/b)/(c/d)  =   (a*d)/(b*d)

From these we can define “reduction”, a.k.a. “reciprocal addition”:
(a/b)   1/+  (c/d)                            =             1   /   (  (1/(a/b) )  +  (1/(c/d)) )
=             lcm(a,c)   /   ( b*(a;d)  + c*(d;a) )

Reduction is like addition, with the roles of numerator and denominator reversed. Therefore they follow similar rules. The following are provable:

(a/A) + (b/B) + (c/C)     =    ( a (lcm(B,C);A)   + b (lcm(C,A);B)  +  c (lcm(A,B);C) )  /  lcm(A,B,C)
(a/A) 1/+ (b/B)1/+ (c/C)     =    lcm(a,b,c)  /  ( A (lcm(b,c);a)   + B (lcm(c,a);b)  +  C (lcm(a,b);c) )

Proof requires these lemmas:
(A;B)*(C; lcm(A,B))      =     (lcm(A,C) ; B)
( lcm(db, dc)  ;  da )     =    ( lcm(b,c)  ;  a  )

Addition and reduction are commutative.
Proof by symmetry of definitions.

Addition and reduction are associative.
Another proof of symmetry of definitions.

Multiplication triple-distributes over addition and reduction.
Proof involves triple-distribution of multiplication over lcm.

Division distributes from the left, and anti-distributes from the right: (trivial proof)
(A+B+C) / D                                       =             (A/D) + (B/D) + (C/D)
(A 1/+ B 1/+C ) / D                    =             (A/D) 1/+  (B/D) 1/+ (C/D)
A / (B+C+D)                                       =             (A/B) 1/+  (A/C) 1/+ (A/D)
A / (B 1/+ C 1/+ D)                    =             (A/B)  +  (A/C)  +  (A/D)

Identities (trivial proof):
(a/b)   =     (a/b) + (0/1)    =   (a/b) 1/+ (1/0)   =   (a/b) * (1/1)

Near-Inverses:
(a/b) + (-a/b)   =   0/b
(a/b) 1/+ (a/-b)   =   a/0
(a/b) * (b/a)    =    (ab)/(ab)    =    1  +   0/(ab)
x/x                        =                            1   +   0/x

0/-1 is known as the “alternator” @; its reciprocal is -1/0, negative infinity.
a/b  + @     =     -a/-b    =     a/b   1/+   - 1/0
Because of them we must weaken distribution to triple distribution:
w(x+y+z)              =             wx  +  wy  +  wz                 if w is finite
w(x 1/+ y 1/+ z)             =             wx  1/+  wy  1/+  wz                    if w is nonzeroid
w(x+y)                  =             wx  +  wy  +  0w                 if w is finite
w(x 1/+ y )         =              wx  1/+  wy 1/+  w/0   if w is nonzeroid

0x * 0y    =   0xy
0x + 0y    =   0(x+y)   =  0(lcm(x,y))
0/(nm)    =    0/n  +   0/(nm)
(nm)/0    =    n/0   1/+   (nm)/0

0/0 is “indefinity”, the indefinite ratio. It is an attractor for addition, reduction, multiplication and division for all finite nonzeroids:
a/b + 0/0   =   a/b 1/+ 0/0   =   a/b * 0/0   =   (a/b)/(0/0)   =   (0/0)/(a/b)    =  0/0
if a and b are both not zero.
Indefinity acts like a generic finite nonzeroid, even though it is both an infinity and a zeroid!
0/0 is an identity for adding infinities, and an attractor for  reducing infinities:
a/0   +  0/0          =             a/b
a/0   +  -a/0         =             0/0
a/0   1/+ 0/0                   =             0/0
0/0 is an identity for reducing zeroids, and an attractor for  adding zeroids:
0/a   1/+  0/0                  =             a/b
0/a   1/+  0/-a                  =             0/0
0/a   + 0/0          =             0/0